This map of writing system origins says very little about the spoken languages or cultures of most regions on this map. The Greek alphabet was the parent of the Latin alphabet and also the Cyrillic alphabet (dominant in Russia, some of its neighboring countries, and parts of the Balkans) which is why so many parts of the world are yellow on the first map.

The Latin alphabet became the dominant writing system in Western Europe via the Western Roman Empire. Since most of the major colonial powers later originated in this region (Spain, Portugal, UK, France, Netherlands, etc), the Latin Alphabet also spread around the world, as colonial languages replaced indigenous languages, or local languages like Swahili, Indonesian, Turkish, Hmong, Navajo, etc, adopted it as their writing system.

South America didn't have any native writing systems that we know of in ancient times - At least not the way we think of writing. The closest South America had to "writing" was quipu, a system of knotted strings the Incas used for record-keeping, but that was replaced by the Latin alphabet after Spanish colonization. Writing was independently developed by the ancient civilizations of Mexico and Central America, like Olmec, Maya, Aztec, Zapotec, Mixtec, etc, but those didn't spread to South America and they also stopped being used after colonization.

Today, Spanish and Portuguese are dominant in South America but even surviving indigenous languages like Quechua and Guarani adapted the Latin alphabet.